Looking for patterns
Most citizen science projects involve large amounts of data than can be boiled down into a visual challenge that requires pattern analysis. The curator of these (and other) projects is Zooniverse, which at latest count has 45 citizen science projects you can get involved in! Such projects are also good for developing your visual search skills, meaning your ability to focus and your visual acuity.
Games
Some new projects are presented as games, such as:
Foldit, a website where you can model the folding of proteins, work that contributes to advances in synthetic chemistry and genetics
Fraxinus, a Facebook game that aids tree fungus research
Play to Cure, an iOS and Android app from Cancer Research.
The Great Brain Experiment, an iOS and Android app that tests your memory, your impulsivity, your attention and decision making, and in so doing not only teaches you about your own cognition, but also enables researchers to get lots of lovely data for telling us about human variation.
EyeWire is an online community of “citizen neuroscientists” who map the retinal connectome (neurons in the retina) by playing an online game.
Lend your computer's power
The third way you can do your bit for science is less direct, and doesn't have the same cognitive benefits for you. You can lend your computer’s power to projects crunching a lot of numbers.
SETI was (I think) the one who began this - they're certainly the best known, at least. SETI@home is a scientific experiment that uses Internet-connected computers in the Search for Extraterrestrial Intelligence (SETI). You can participate by running a free program that downloads and analyzes radio telescope data.